Idempotency and monotonicity of entailment are inference rules that directly operate on the judgments or the relations between antecedents and consequents. Idempotency of entailment states that the same consequences can be derived from many instances of a hypothesis as just from one (“A, B, B ⊢ C” can be contracted to “A, B ⊢ C” leaving the entailed consequence C intact). Monotonicity of entailment, on the other hand, means that the hypotheses of any derived fact can be arbitrarily extended with additional assumptions (“A ⊢ C” can be assumed as “A, d ⊢ C” where d is the additional assumption and C is the unchanged consequence). Here, the turnstile symbol ⊢ denotes entailing. Antecedents are on the left-hand side of the turnstile,and consequents on the right-hand side. Idempotency of entailment implies the availability of antecedents as free resources (in the context of reasoning via artifacts, different instances of application or use for a given artifact do not change the outcome). And monotonicity of entailment implies context-independency of reasoning (extending the role of an artifact or adding new assumptions about its use in bringing about some ends does not alter the result).
For introductions to the philosophies of ancient Cynicism, Stoicism, and Confucianism, see: William Desmond, Cynics (Stocksfield: Acumen, 2006); John Sellars, The Art of Living: The Stoics on the Nature and Function of Philosophy (Bristol: Bristol Classical Press, 2009); Philip J. Ivanhoe, Confucian Moral Self Cultivation (Indianapolis: Hackett Publishing Company, 2000).
Research on computational dualities and concurrency can be traced back to the works of Marshall Stone and Carl Adam Petri. Stone’s application of mathematical dualities (bijective correspondence between sets and equivalence relations between categories as inverse functors) to Boolean algebra set up a framework for a deeper analysis of the semantics of information processing. Petri’s contributions to computer science—most notably his Petri nets, which were originally invented to describe chemical processes—provided the necessary modeling tools for studying process execution and problems associated with concurrent computation, such as scheduling and resource management (see the “dining philosophers” problem). But the main breakthroughs in the study of computational dualities have only been made recently through the intersection of different lines of research on asynchronic models of concurrency in physical systems (see, for example, the work of Peter Wegner), mathematical and computational models of nonsequential interaction games (see Robin Milner, Andreas Blass, and Samson Abramsky), and substructural logics and proof theory, particularly the work of Jean-Yves Girard.
To be continued in “What Is Philosophy? Part II: Programs and Realizabilities”
All images: “The Study of Hidden Symmetries in Raphael’s The School of Athens,” from Guerino Mazzola, Detlef Krömker, and Georg Rainer Hofmann, Rasterbild — Bildraster (Anwendung der Graphischen Datenverarbeitung zur geometrischen Analyse eines Meisterwerks der Renaissance: Raffaels “Schule von Athen”)